Mathematics > Geometric Topology

Title:
Analogies between group actions on 3-manifolds and number fields

Abstract: Mazur, Kapranov, Reznikov, and others developed ``Arithmetic Topology,'' a
theory describing some surprising analogies between 3-dimensional topology and
number theory, which can be summarized by saying that knots are like prime
numbers. We extend their work by proving several formulas concerning branched
coverings of 3-manifolds and extensions of number fields and observe that these
formulas are almost identical, via the dictionary of arithmetic topology. Until
now there is no satisfactory explanation for the coincidences between our
formulas. The proofs of topological results use equivariant cohomology and the
Leray-Serre spectral sequence. The number theoretic proofs are based on an
approach to class field theory via idele groups.

Comments:

21 pages, an extended version of a paper to appear in Comm. Math. Helv